John Baez's Description of the Millenium Prize Problems

There is a 1 million dollar prize for the solution to each problem offered
by the Clay Mathematics Institute. See http://www.claymath.org
for more details. Here are John Baez's descriptions of the problems:

1. P = NP?

This is the newest problem on the list and the easiest
to explain. An algorithm is "polynomial-time" if the time it takes to run
is bounded by some polynomial in the length of the input data. This is
a crude but easily understood condition to decide whether an algorithm
is fast enough to be worth bothering with. A "nondeterministic polynomial-time"
algorithm is one that can check a purported solution to a problem in
an amount of time bounded by some polynomial in the input data. All algorithms
in P are in NP, but how about the converse? Is P = NP? Stephen Cook posed
this problem in 1971 and it's still open. It seems unlikely to be true
- a good candidate for a counterexample is the problem of factoring integers
- but nobody has proved that it's false. This is the most practical
question
of the lot, because if the answer were "yes", there's a chance that one
could use this result to quickly crack all the current best encryption
schemes.

2. The Poincaré conjecture.

Spheres are among the most fundamental topological spaces, but spheres
hold many mysteries. For example: is every 3-dimensional manifold with
the same homotopy type as a 3-sphere actually homeomorphic to a 3-sphere?
Or for short: are homotopy 3-spheres really 3-spheres? Poincaré
posed this
puzzle in 1904 shortly after he knocked down an easier conjecture of his
by finding 3-manifolds with the same homology groups as 3-spheres that
weren't really 3-spheres. The higher-dimensional analogues of Poincaré's
question have all been settled in the affirmative - Smale, Stallings and
Wallace solved it in dimensions 5 and higher, and Freedman later solved
the subtler 4-dimensional case - but the 3-dimensional case is still unsolved.
This is an excellent illustration of a fact that may seem surprising at
first: many problems in topology are toughest in fairly low dimensions!
The reason is that there's less "maneuvering room". The last couple decades
have seen a burst of new ideas in low-dimensional topology - this has been
a theme of This Week's Finds ever since it started - but the
Poincaré conjecture
remains uncracked.

3. The Birch-Swinnerton-Dyer conjecture.

This is a conjecture about elliptic curves, and indirectly, number theory.
For a precise definition of an elliptic curve I'll refer you to "week13"
and "week125", but basically, it's a torus-shaped surface described by
an algebraic equation like this:

y2 = x3 + ax + b

Any elliptic curve
is naturally an abelian group, and the points on it with rational coordinates
form a finitely generated subgroup. When are there infinitely many such
rational points? In 1965, Birch and Swinnerton-Dyer conjectured a criterion
involving something called the "L-function" of the elliptic curve. The
L-function L(s) is an elegant encoding of how many solutions there are
to the above equation modulo p, where p is any prime. The Birch-Swinnerton-Dyer
conjecture says that L(1) = 0 if and only if the elliptic curve has infinitely
many rational points. More generally, it says that the order of the zero
of L(s) at s = 1 equals the rank of the group of rational points on the
elliptic curve (that is, the rank of the free abelian summand of this group.)
A solution to this conjecture would shed a lot of light on Diophantine
equations, one of which goes back to at least the 10th century - namely,
the problem of finding which integers appear as the areas of right triangles
all of whose sides have lengths equal to rational numbers.

4. The Hodge conjecture.

This question is about algebraic geometry and topology. A "projective nonsingular
complex algebraic variety" is basically a compact smooth manifold described
by a bunch of homogeneous complex polynomial equations. Such a variety
always has even dimension, say 2n. We can take the DeRham cohomology of
such a variety and break it up into parts Hp,q
labelled by pairs (p,q)
of integers between 0 and n, using the fact that every function is a sum
of a holomorphic and an antiholomorphic part. Sitting inside the DeRham
cohomology is the rational cohomology, The rational guys inside
Hp,p
are called "Hodge forms". By Poincaré duality any closed analytic subspace
of our variety defines a Hodge form - this sort of Hodge form is called
an algebraic cycle. The Hodge conjecture, posed in 1950 states: every Hodge
form is a rational linear linear combination of algebraic cycles. It's
saying that we can concretely realize a bunch of cohomology classes using
closed analytic subspaces sitting inside our variety.

5. Existence and mass gap for Yang-Mills theory.

One of the great open problems of modern mathematical physics is whether
the Standard Model of particle physics is mathematically consistent. It's
not even known whether "pure" Yang-Mills theory - uncoupled to fermions
or the Higgs - is a well-defined quantum field theory with reasonable properties.
To make this question precise, people have formulated various axioms for
a quantum field theory, like the so-called "Haag-Kastler axioms". The job
of constructive quantum field theory is to mathematically study questions
like whether we can construct Yang-Mills theory in such a way that it satisfies
these axioms. But one really wants to know more: at the very least, existence
of Yang-Mills theory coupled to fermions, together with a "mass gap" -
i.e., a nonzero minimum mass for the particles formed as bound states of
the theory (like protons are bound states of quarks).

6. Existence and smoothness for the Navier-Stokes equations.

The Navier-Stokes equations are a set of partial differential equations
describing the flow of a viscous incompressible fluid. If you start out
with a nice smooth vector field describing the flow of some fluid, it will
often get complicated and twisty as turbulence develops. Nobody knows whether
the solution exists for all time, or whether it develops singularities
and becomes undefined after a while! In fact, numerical evidence hints
at the contrary. So one would like to know whether solutions exist for
all time and remain smooth - or at least find conditions under which this
is the case. Of course, the Navier-Stokes equations are only an approximation
to the actual behavior of fluids, since it idealizes them as a continuum
when they are actually made of molecules. But it's important to understand
whether and how the continuum approximation breaks down as turbulence develops.

7. The Riemann hypothesis.

For Re(s) > 1 the Riemann zeta function is defined by

zeta(s) = 1/1s + 1/2s + 1/3s + ....

But we can extend it by analytic continuation to most
of the complex plane - it has a pole at s = 1.
The zeta function has a bunch of zeros in the "critical strip" where
Re(s) is between 0 and 1. In 1859, Riemann conjectured that all such zeros
have real part equal to 1/2. This conjecture has lots of interesting
ramifications
for things like the distribution of prime numbers. By now, more than a
billion zeros in the critical strip have been found to have real part 1/2;
it has also been shown that "most" such zeros have this property, but the
Riemann hypothesis remains open.

If you solve one of these conjectures and win a million dollars
because you read about it here on This Week's Finds, please put me in your
will.

Previous issues of "This Week's Finds" and other expository articles
on mathematics and physics, as well as some of Baez's research papers,
can be obtained at http://math.ucr.edu/home/baez/